Exact Statistical Solution of Pseudorange Equations

摘要

Although the exact GPS solution proposed by Bancroft is nonlinear, it may be manipulated into a linear form when 5 or more satellites are visible. This linear form is exact, as opposed to the linear solution obtained via repeated linearization in the iterated least squares (ILS) method. By virtue of this exactness, the solution of the linear form is always the true user position, while the ILS may converge to an incorrect solution (this is especially common when the GPS user is in space). When the measured pseudoranges are noisy, the linear structure ensures that the position estimate will converge to the correct value and that the error covariance of the estimate is known, guarantees that have not been found for nonlinear estimators that use the Bancroft solution directly. The conversion to the linear form excludes information present in a single scalar nonlinear measurement equation. We demonstrate several procedures for refining the linear estimate with this remaining information. In addition, we show that the methodology developed for direct GPS solutions can be applied to create linear direct methods for differential GPS problems.